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Ultraelliptic Integrals and Two-Dimensional Sigma Functions

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Abstract

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u) = 0}, where σ(u) is the two-dimensional sigma function. We show that G(z) = F(ξ(z)), where z is a local coordinate in a neighborhood of a point of the smooth curve W and ξ(z) is the smooth function in this neighborhood given by the equation σ(u1, ξ(u1)) = 0. We obtain differential equations for the functions F(z), G(z), and ξ(z), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function G(z) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.

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Acknowledgments

The authors are grateful to A. B. Bogatyrev and V. Z. Enolski† for stimulating discussions and literature references.

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Authors and Affiliations

  1. Advanced Mathematical Institute, Osaka, Japan

    T. Ayano

  2. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    V. M. Buchstaber

Authors
  1. T. Ayano
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  2. V. M. Buchstaber
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Correspondence to T. Ayano or V. M. Buchstaber.

Additional information

To the memory of the remarkable mathematician I. M. Dektyarev (1940–2002)

Russian Text © The Author (s), 2019. Published in Funktsional’nyi Analiz i Ego Prilozheniya, 2019, Vol. 53, No. 3, pp.3–22.

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Ayano, T., Buchstaber, V.M. Ultraelliptic Integrals and Two-Dimensional Sigma Functions. Funct Anal Its Appl 53, 157–173 (2019). https://doi.org/10.1134/S0016266319030018

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